Question 1

What is a function in mathematics?

Accepted Answer

A function is a relation that assigns exactly one output value to each input value. It is often written f(x), read "f of x", where x is the input and f(x) is the output. The set of valid inputs is the domain; the set of possible outputs is the range. For example, f(x) = 3x − 1 takes any real number x and produces a unique output: f(2) = 5, f(0) = −1, f(−3) = −10. A relation is not a function if any single input maps to more than one output.

Question 2

How do you evaluate f(a)?

Accepted Answer

To evaluate f(a), replace every occurrence of x in the function rule with the value a, then simplify. For f(x) = x² − 4x + 7: f(3) = (3)² − 4(3) + 7 = 9 − 12 + 7 = 4. For f(x) = 2x + 1: f(−5) = 2(−5) + 1 = −10 + 1 = −9. When evaluating at an algebraic expression like f(a + h), substitute x = a + h and expand: f(a+h) = (a+h)² − 4(a+h) + 7 = a² + 2ah + h² − 4a − 4h + 7.

Question 3

What is function composition and how do you compute it?

Accepted Answer

Function composition (f∘g)(x) = f(g(x)) means apply g first, use its output as the input for f. Example: g(x) = x + 2, f(x) = x². Then f(g(x)) = f(x+2) = (x+2)² = x² + 4x + 4. To evaluate at a point: (f∘g)(3) = f(g(3)) = f(5) = 25. The composition is defined only when the output of g lies in the domain of f.

Question 4

Is f(g(x)) always equal to g(f(x))?

Accepted Answer

No. Function composition is generally not commutative. f(g(x)) and g(f(x)) are equal only in special cases. For f(x) = x + 1 and g(x) = 2x: f(g(x)) = f(2x) = 2x + 1, while g(f(x)) = g(x+1) = 2(x+1) = 2x + 2. These differ by 1 for every x. Exceptions include f and g being inverses of each other (f(g(x)) = g(f(x)) = x) or both being the identity function.

Question 5

What are the sum, difference, product, and quotient of functions?

Accepted Answer

Given functions f and g, you can create new functions: (f+g)(x) = f(x) + g(x); (f−g)(x) = f(x) − g(x); (f·g)(x) = f(x) · g(x); (f/g)(x) = f(x)/g(x), where g(x) ≠ 0. Example: f(x) = x² and g(x) = x − 1. Then (f+g)(x) = x² + x − 1; (f·g)(x) = x²(x−1) = x³ − x²; (f/g)(x) = x²/(x−1), domain excludes x = 1.

Function f(x)	Evaluate at	Substitution	f(x) =
x² − 3x + 2	x = 4	(4)² − 3(4) + 2 = 16 − 12 + 2	6
2x + 5	x = −3	2(−3) + 5 = −6 + 5	−1
x³ − 1	x = 2	(2)³ − 1 = 8 − 1	7
3x² − 2x + 1	x = 0	3(0)² − 2(0) + 1	1
(x+1)²	x = 5	(5+1)² = (6)²	36
−x² + 4	x = 3	−(3)² + 4 = −9 + 4	−5

f(x)	g(x)	f(g(x))	g(f(x))	Equal?
x + 1	2x	2x + 1	2x + 2	No
3x	x − 2	3x − 6	3x − 2	No
x²	√x	x (for x≥0)	x (for x≥0)	Yes (inverses)
eˣ	ln(x)	x (for x>0)	x (for all)	Yes (inverses)
2x + 3	(x−3)/2	x	x	Yes (inverses)

Function Calculator

Formula

Function Calculator

Function Evaluation Examples

Function Composition Examples

Related Calculators

Disclaimer

Frequently Asked Questions